PLASMA DYNAMICS
XI.
Prof. S. C. Brown
Prof. G. Bekefi
Prof. D. R. Whitehouse
M. L. Andrews
V. Arunasalam
C. D. Buntschuh
J. F. Clarke
J. D. Coccoli
A.
F.
E.
W.
E.
J.
P.
R.
PLASMA PHYSICS*
X.
W.
H.
B.
C.
W.
L.
D.
J.
W.
J.
H.
G.
F.
R.
Crist
Fitzgerald, Jr.
Glenn, Jr.
Hooper, Jr.
Ingraham
Jameson
Kronquist
T. Llewellyn-Jones
J. McCarthy
J. Mulligan
J. Nolan, Jr.
R. Radoski
L. Rogoff
Y-F. Tse
E. Whitney
EXCITATION OF THE OPTICAL SPECTRUM IN THE HOLLOW-CATHODE ARC
1. Introduction
The intensity of a spectral line from a unit volume of plasma, with self-absorption
and broadening neglected, is given by
Ijk = nAjkhjk
(1)
,
.th
where n. is the density of emitting atoms or ions in the j
excited state, Ajk is the tran-
sition probability from the jth state to the kth state, and hvjk is the photon energy. In
a low-pressure arc plasma, which is far from thermal equilibrium, it is necessary to
consider the rates of the various populating processes in order to arrive at an expression for n. in terms of the plasma properties.
1
J
The base pressure
The plasma that we are studying is a highly ionized argon arc.
13-3
before ionization is a few 4 Hg. The electron density is of the order of 1013 cm ; electron temperatures are several volts at the center of the column.
Ion temperatures may
be one volt, or more.
The spectrum from 3200 A to 5200 A contains, for the most part,
AII lines and several weak AI and AIII lines.
In a plasma the population process is frequently predominantly electronic excitation
from the ground state, and the depopulation process is principally spontaneous emission
to all lower states.
n
p .n
03 o
n. J
where
.ois
In this case,
-
A.
(2)
the excitation coefficient, n o is the density of atoms or ions in the ground
state, and A. is the transition probability per second to all lower states.
We shall use
*This work was supported in part by the U. S. Atomic Energy Commission (Contract
AT(30-1)-1842); and in part by the U. S. Air Force (Electronic Systems Division) under
Contract AF19(604)-5992.
QPR No. 70
(XI.
PLASMA PHYSICS)
this expression in analyzing the data; however, for the conditions obtaining in the arc,
radiative recombination, stepwise electronic excitation, and cascading may give significant contributions to n..
In this report we consider these other processes and estimate
their rates relative to electronic excitation.
We shall find that the competing processes
may have rates comparable to, but somewhat smaller than, electronic excitation, so
that Eq. 2 represents a reasonably good approximation.
In plasmas at micron pressures there is negligible self-absorption of nonresonance
radiation at optical frequencies, and negligible population by collisions of the second
kind.
We shall also neglect the effects of metastables.
for the argon atom.
This is probably not legitimate
If the ion has any metastable levels, they are probably close enough
to the radiating levels (less than T ) to be considered as merged into a single radiating
state.
Throughout this report a Maxwellian electron distribution is assumed.
This is
reasonable in view of the high electron temperatures and high degree of ionization.
Because none of the relevant cross sections and atomic constants are available for the
argon ion, and very few for the atom, it has been necessary to make order-of-magnitude
guesses of the values.
2.
Excitation Coefficients
The coefficient Pij for excitation from the it h to the jth state by electron impact is
the average over the electron distribution function of the cross section a.. multiplied
by the electron velocity v.
8eT
rrm
Pij :
For a Maxwellian at temperature T
-
-u/T_
x/T- a.-(u) y
e
volts,
u
d
,
(3)
where u is the electron energy, and Vx = Vj - V i is the difference in excitation potential. The ionization coefficient po0 is also given by Eq. 3.
There are very limited data on the cross sections, especially for argon. There are
no data for the excitation or ionization from the ground state of the ion, or from higher
levels of either the atom or ion. We can obtain a useful approximation by noting that
all of the cross sections have the general form shown by the solid line in Fig. XI-1.
For excitation, am is of the order of 10 -19 cm 2 , and Vm is slightly greater than Vx
For ionization, om ~ 10
-16
2
cm , and V m is several times V 0, the ionization potential.
Three approximations to u-(u) which permit Eq. 3 to be readily integrated are shown as
dotted lines:
al
=
C1 /U
2 = 2(-V
'3 =
QPR No. 70
m
u >V
(XI.
a(n)
PLASMA PHYSICS)
I
I
,fix F
°'m
---
--
Su
V
V
x
m
ELECTRON ENERGY
Fig. XI-1.
For (l,
we use the theoretical formula
a- = C(ln u)/u, suppressing the slowly varying
a" is the popular approximation for low T; a-3 improves on this for higher
logarithm.
T.
Approximate excitation and ionization cross sections.
The resulting coefficients, in mks units, are
p1
=
,//2
-
e
mx
-V/T
)T-1/2 e Vx/T
2T
8e
V
3
-1/2
8e
wm T
C 11
8e
8e
+
m
V
T
- 1/2 e
T
x /T_
At low electron temperatures the exponent contains the principal variation of p with
T , and it makes little difference which coefficient is used. At high temperatures, the
first expression, which takes into account the decreasing cross section at high energy,
is presumably the best approximation. For this reason, we use Pli, choosing C l = UmV
For argon atoms, V.
o
ij
e VxT cm 3 /sec.
T
1. 1 X 1O0 -10-1/2
o
PoO
= 15. 8 volts, we have
-7
= 1. 1 X 10
For argon ions, V.
QPR No. 70
-1/
T
e
-15. 8/T
/T
= 27. 5 volts, we have
(4a)
(4b)
.
(XI.
PLASMA PHYSICS)
P ij
2/
= 1.8 X 10- 1 0 T
+
e
(4c)
-2
1. 8X10-7T-1/2
7
. 5/T
(4d)
The average cross sections for excitation and ionization of an atom already in an
excited state (for example,
the resonance level) seem to be an order of magnitude
greater than those of an atom in the ground state.3
For the most part,
this is due to
the fact that V. - V. is much smaller than V. - V . Possibly there is an increase in
J
1
j
o
am because of the greater size of the excited atom, but for order-of-magnitude estimates we shall ignore this possibility.
3.
Densities of Singly and Doubly Ionized Ions
It is necessary to know the densities of neutrals, singly and doubly ionized ions, and
electrons in order to compute the rate of excitation from the ground state and rate of
volume recombination.
To estimate these densities,
consider the following model.
atom has first and second ionization potentials V 1 and V 2,
excitation and higher degrees of ionization.
electron impact from the ground state,
p , respectively.
coefficients po and
a time constant
respectively.
The
Neglect
Suppose that ions are generated only by
that is,
A -
A+ and A+ -
A++,
with ionization
Assume that recombination is at the walls,
with
for either ion.
T
In the steady state, the particle balance equations are
dn
dto
0 = -1 (n +n
dt
7
n
ponn
+
-
=
dt
dnd++
n
o
+Pn_) n
(5)
Tn
d-0
dt
o
) -
++
=n
+
n+ --
- +
1
7
n.
++
The total heavy-particle concentration N = n o + n
requires n_ = n+ + 2n ++.
n
n
oIS
k1 +k 2
I k2
2Nklk2
k +k
1 2 +1
2Nkk2
L
N-
Solving Eqs. 5,
+ n++ is
1+k n
n+
N
(
+kn
(+k
(I
QPR No. 70
kn
charge neutrality
subject to these constraints, yields
Nk -1
N2 k1 l k 2
kln
1
N
fixed;
(6)
)(lkn
n-) (1 +k1 n_)
klk n 2
( +n_)(l
+2kln
n
N
(1+k n_)(1+k
_)
1 n_)
(XI.
k
where k I = pOT,
Z
PLASMA PHYSICS)
= P+T.
The excitation coefficients are given in Eq. 4b and 4d. The mean loss time for ions
is difficult to evaluate. It must be somewhere between the free-flight loss time to the
In the
ends of the arc column and the radial diffusion time across the magnetic field.
arc, this is between 10
S= 10 -
3
and 10 -
4
5
and 10 -
2
sec.
sec, respectively,
In Fig. XI-2, n o , n+, and n++ are plotted for
for N = 1014 cm .
If triply ionized atoms were
the n++ density would reach a maximum and then fall off,
included,
build up as T
and n+++ would
It is important to note that for T_ just roughly 0. 5 volt above
increases.
n++/n+ is 5-15 per cent, which value is an appreciable fraction of the total
threshold,
ionization.
The actual operating point of the arc can be varied by adjusting the arc current,
gas-flow rate, and magnetic field. It should also be remembered that the plasma is
also generated inside the hollow cathode where the density and temperature are much
higher. It is, therefore, quite possible that the electron temperature in the positive
column is somewhat lower than that calculated on this model for a given electron density.
4.
Imprisonment of Resonance Radiation
If resonance radiation is strongly absorbed by atoms or ions in the ground state, it
is imprisoned and leads to increased population of the resonance levels. Here we take
the resonance levels to be the lower excited states, and neglect the higher states that
have alternate transitions available.
To estimate the degree of imprisonment, let the
excitation of a resonance level be by electron impact and photon absorption,
and let
de-excitation by all processes except spontaneous emission have a time constant Td.
Then, the balance equations for atoms in the resonance level and photons are
n
dn
-d
dt
dn
ol
0 =
n
S0 = An
dt
0
Bpn+0
+ Bpn
--
Bpn
o
1
T
d
An
(7)
I
1
n
(8)
7
where A and B are the Einstein coefficients,
p is the monochromatic
for resonance radiation, n 1 is the resonance level population, n
sity in the line, and
T
is the photon escape time.
energy density
is the total photon den-
We shall approximate
p by n hv/AvD
ZkT
where AvD=
/
is the Doppler width of the line.
Holstein 4 has given an expression for
T
in the limit of large absorption for cylin-
drical geometry and Doppler-broadened radiation:
QPR No. 70
10x 10 13
10x 1013
9
8
7
6
2.5
3
3.5
4
4.5
5
5.5
2
2.5
ELECTRON TEMPERATURE (VOLTS)
Particle densities in highly ionized plasmas.
(b) Ion loss time T = 10 -4 sec.
4.5
4
3.5
ELECTRON TEMPERATURE
(a)
Fig. XI-Z.
3
(VOLTS)
(b)
(a)
Ion loss time T = 10
-3
sec.
PLASMA PHYSICS)
(XI.
k R JFTrInk R
o
(9)
o
1. 6A
Tv
where
k
2ZAg
n
o
1
c
1
8r 3/2 v2 g 0 AvD
o
(10)
is the absorption coefficient at the center of the line.
If we introduce into Eq. 7 an imprisonment factor defined by the ratio of absorption
to emission, J = Bpno/An 1 , we can see that the density n 1 is governed by spontaneous
emission, or other processes accordingly as (1-J)ATd is much greater or much less
From Eqs.
than one.
we obtain
8-10 and the ratio of Einstein coefficients,
1
(11)
T
TO
1+c
n
sec
volt-1 for both neutrals and ions, if we take A = 5 X 107
8
-1
- I
= 700 A for
, gog 1 = 1,
= 1/4, X = 1050 A for the atom, and A = 6.6 x 10 sec
where c = 2. 3
go/g
2
o
1024 cm
the ion (rather arbitrarily),
and R = 3 cm,
J1n kR
= 2. The transition probabilities
are uncertain, but are probably of the correct order of magnitude.
for ions in the arc is roughly bracketed by 9 X 10-5 (T+
The value of 1 - J
n=
10
ulating
. i vv0 n
2 (T+ = 0.
and 4.-5 X
13 cm)
1 , n
(T
cm) and 4. 5 X 10
= 5 X 10
13
3
The other principal depop-
cm ).
superelastic or
-5
-4
sec.
For these, Td may be between 104 and 10
processes
electrons.
are
likely to
be
= 2,
either
ionizing collisions
with
Thus we might expect
(l-J)ATd to be between 0.6 and 3 X 103. This means that the imprisonment is generally not quite great enough for us to neglect spontaneous emission as a depopulating
process as compared with the other losses, but it is enough to decrease the effective
transition probability by several orders of magnitude.
5.
Competing Processes
We now consider the relative rates of excitation and de-excitation of a nonresonant
level. Neglect photon absorption and populating collisions of the second kind. Assume
that stepwise excitation is from the resonant level only, and that cascading is from a
single upper level.
The balance equation is
dn.
0 =
ojn-no +
jn_nl + Akjn k + ajnon
- n [A +Td
QPR No.
70
jkn_+jn_,
c
j
]
(12)
(XI.
PLASMA PHYSICS)
where n o , n l , n , nk are the populations of the ground, resonant, radiating, and cascading levels, respectively; no is the ground-state density of ions of next higher degree;
are the mean times for escape from
a. is the recombination coefficient; and Td and T
J
J
Cj
the plasma and collisions of the second kind, respectively. Typical values of all coefv
ficients are given in Table XI-1 for the argon ion, with V1 = 16v , V. = 20 , Vk = 2ZZ
vV
and V,
= 27. 5
It is quite apparent that the upper levels de-excite almost exclusively by spontaneous
emission, as expected.
Radiative recombination is the only significant volume recombination process in the
arc.
The partial coefficient for recombination to an individual level decreases rapidly
with distance above the ground state, especially when the electron energy is high. 5 The
formula a. = 10
T13T-3/4 represents only a crude estimate based on calculations for mon-
atomic hydrogen. 3 '
5
In Table XI-2 we give the ratio of recombination rate to electron
excitation rate, using the formulas in Table XI-1, and taking n++/n+ from Fig. XI-2a.
It is seen to be a marginal proposition to neglect this process, especially at high arc
currents, when n+ begins to decrease.
We can obtain an upper limit to the ratio of electron excitation from the resonance
level to that from the ground state by assuming 100 per cent imprisonment and depopulation of the resonance level by ionization by electron impact.
j~nl
S.n
Pljp
-3
- 10
-
ojno
e
This ratio is
(Vo -V1)/T_
oj 100
(see Table XI-2).
Incomplete imprisonment and other resonance level losses - particu-
larly superelastic collisions - tend to decrease this ratio.
If we assume that the cascade level is populated only by electron excitation from the
ground state and depopulated only by spontaneous emission, the ratio of cascade to excitation contributions to n. is simply
Akjnk
Akj
ojnno
Ak
-(Vk-V.)/T
Now, Akj is proportional to (Vk-V) 3 , so we would expect the branching ratio Akj/Ak
Thus we let Vk - V. =
to be close to one only for transitions at optical frequencies.
2v
in the comparison in Table XI-2.
Again, we have a marginal situation, but the esti-
mates are presumably on the high side.
6.
Intensity Measurements
A Jarrell-Ash 0. 5-meter scanning monochromator was used in measuring the inten-
sities of the argon ion lines.
QPR No. 70
In the initial survey runs, light from the entire arc was
Table XI-I.
Values of population coefficients.
6v
Coefficient
10
cm
10-
10
-4/T_
T - 1/2 e4/T
10
1.8X =110
10
Pj= 1. 8X
pjk
=
-20/T
T-/2
1. 8 X 10p0 = p=
1.8X
1 3
a.= I0
1
S10
e
-7.
-104
4
sec
6. 1 X 10 -13
2. 6X 10
7. 1
3. 31X 10
3. 8X 10 - 11
3. 0
cm3/sec
~ 109 sec 2
5/T_
10I12
10
6.0% 10
-11
-9
-14
4.3
4.5 X 10 8
X 10
10-
8
1. 4X 10
3.5
X10
2.I
-14
1
4
.
~ 104 sec
Table XI-2.
1
-1
-1
Ratios of secondary processes to electronic excitation.
Ratio
T
ajnoo
Pojno
20/T
S5.5 X 10-
4
T - 1/4 e
ojno
e
3
n
++
n+
.097
.24
.60
.32
.12
.046
.37
.48
.51
o
Akjn k
p .n n
QPR No. 70
2. 4
n
i 10
0]
=2
-2/T
= e
10--8
2.6X 10
Tdj
T
-12
5. 8X 10 "
2.8X 10
-7 5T
- 1/ 2
T
Akj, A
/sec
T-I/2 e 3/T
-1/
= 1.8X
10-1310-7
3T T
3
-14
(XI.
PLASMA PHYSICS)
allowed to fall on the entrance slit,
2 feet from, the arc axis.
which was perpendicular
to,
and approximately
Later, a system of collimating slits and mirrors was inter-
posed between the arc and the monochromator.
'
--
This arrangement, Fig. XI-3,
takes a
AX IS
I
MIRROR
I
I-~
SENTRANCE
TO
SLIT
I
SLIT
SLIT
MIRROR
Fig. XI-3.
Movable collimator.
narrow ribbon of light (0. 005 in.) parallel to the arc axis and rotates it 90' so that it
falls on the monochromator entrance slit.
A lens was used to focus the light on the slit.
The collimating-slit system is movable, scanning the arc perpendicular to the axis.
provision has been made for absolute intensity measurements;
arbitrary units.
all measurements are in
A tungsten-ribbon filament lamp is being added to a new arrangement
for calibration of intensities.
The 1P21 photomultiplier output was amplified by a
Philbrick UPA-Z operational amplifier,
recorder.
and displayed on a Varian G11A strip chart
Often the intensity variation in one run would be large enough to necessitate
changing the amplifier gain or photomultiplier voltage to keep the recorder on scale.
these cases, the run was made in overlapping segments,
on log paper.
The gain factor was obtained from the displacement of the curves.
Light from the Whole Arc
(i)
(ii)
A spectrogram from 3200 A to 5200 A.
Intensity versus are current for several ion lines.
(iii) Intensity versus magnetic field for several ion lines.
(b)
Ribbon of Light (Arc Profiles)
(i)
QPR No. 70
In
and the raw data were plotted
The measurements performed thus far have been on
(a)
No
Arc profiles of several ion lines and many are conditions.
(XI.
(ii)
PLASMA PHYSICS)
Intensity on axis versus current.
A discussion of these measurements follows.
Approximately 30 AI, 190 All, and 7 AIII lines have been identified out of more
than 300 argon lines in a spectrogram taken at 20 amps and 800 gauss. All of the AI
and AIII lines are rather weak compared with the most intense lines of the single ion.
a. i.
Twelve, out of 40, unidentified impurity lines had measurable, but very small, intensities. They are possibly Ta, Ne, or Fe lines, but the wavelengths are not precise
for us to
enough
intensity of the
The
be sure.
Ta 4580. 7 ?) is approximately 10
-3
- 3
strongest
impurity line
(k4580;
times the intensity of the stronges AII line (k 4348).
Ion lines, strong enough to be of use, are listed below in order of increasing excitation potential.
a. ii.
I (arb)
X
Vx
x
I (arb)
V
19. 14
4806
25, 000
19. 88
3851
9, 550
19. 18
4736
22, 800
21. 05
4609
16, 600
19. 41
4348
25, 000
21. 40
4370
11, 800
19. 46
4426
25, 000
22. 85
3588
5, 400
19. 88
3729
9, 800
23. 07
3868
1, 670
The intensities of X4806,
those of X4370,
X4736,
x
X4348, X4426 are plotted in Fig. XI-4 as Ii,
X3588 as 12I all normalized to 100 at 65 amps.
The magnetic field was
The flow rate was 130 cc-atm/min which produced a pressure of 2-3 a Hg
in the arc tube. Although there is considerable scatter in the data points, there is a distinct difference between the intensities of lines originating at levels between 19. 1 volts
1200 gauss.
and 19. 5 volts and those originating between 21. 4 volts and 22. 9 volts.
If we assume that the line intensities are given by
Cin n+T -1/2
I. =I c.n
-Vj/T
j = 1, 2,
and take logarithms of intensity ratios, we can see how T_ varies once it is known for
Taking ratios at
V 2 - V 1 is approximately equal to 3 volts.
one current setting.
30,
65, and 100 amps, and assigning values to T 100, we obtain the values listed below
QPR No. 70
'30
T6
1.9
2.2
2. 4
2. 1
2.4
2.6
2.2
2.5
2. 8
2.3
2. 8
3. O0
2. 5
3.2
3. 5
2. 9
3. 6
4.0
5
(XI.
PLASMA PHYSICS)
160
140 -
LINE INTENSITIES vs ARC CURRENT
B
1200 GAUSS
F = 125 - 130 cc-atm/min
120 -
100
z
80
ALL INTENSITIES NORMALIZED AT 65AMPS
_
60-
11-
z
40
20
0
0
10
20
30
40
50
60
70
I
I
90
80
100
110
ARC CURRENT (AMPS)
Fig. XI-4.
Line intensities vs arc current.
The fact that the intensity of ion lines increases linearly with current is rather
surprising, since it is proportional to electron density times ion ground-state density. Since, even at 30 amps, the concentration of n++ is great enough to produce
significant AIII line intensity,
it
is probable that the arc is
operating near the peak
of the n+ density.
The slower increase in intensity at currents higher than 80 amps
could then be explained by a depletion in n+ density as n++ builds up; comparison
of this table with Fig. XI-2 supports this hypothesis.
a. iii.
All line intensities increased gradually as the magnetic field was increased
from 1000 gauss to 2200 gauss, at 65 amps arc current.
Total increase was appoximately a factor of 2, and no significant dependence on excitation potential could be
discerned.
b. i.
A typical profile is shown in Fig. XI-5.
1200 gauss,
and 100 cc-atm/min (1-1. 5
This curve is
X4348 at 50 amps,
4 Hg in arc tube).
The original idea in
this experiment was to measure the radial variation of intensity for three lines with
widely separated upper levels, for example, X4348, X4370, X3868 at 19. 4, 21. 4,
23. 1 volts,
and deduce the radial variation in electron temperature,
by comparing
intensity ratios as we did in (a.ii). This experiment has not yielded useful results
because accurate alignment of the monochromator could not be maintained because
QPR No. 70
100
PLASMA PHYSICS)
(XI.
X = 4348X
B = 1200 GAUSS
I
F
-
sweep
which should
through
plasma density profile,
we shall
be able
as
not radius,
enough
from probe measurements,
temperature
close
to the
intensity
curve
to
line
single
are
worth
edge
the
of the
tem-
obtain the
noting:
The
fusion.
increase
is
variation of
several
for
of
a few
a
much
increased,
but no
width
(reciprocal
in
Fig. XI-6.
conditions
per
width of the
of the
cent
in
width as
more pronounced
systematic
B
increase
measurements
The width of the crown--the
portion
of the
If
101
slope)
anything,
runs
from
as
current
have
from
not determined solely by the cathode diameter.
QPR No. 70
column,
arc
axis -
from arc
The slope of the log I curve
not yet been inverted.
have
the data
a measure
then affords
is
use a
electron
being built
now
is
probe
run.
the
to solve
been made
The intensity falls off almost exponentially with distance
(i)
(ii)
to
the
run to
to enable density and other probe
Once we know,
up.
of Fig. XI-3
Two features
There
and
has
constant
A moving
again.
fast
from
distance
profile.
perature
shown
the arc
without burning
measurements
arc,
shall try
and we
problem,
0.9
(INCHES)
mount
and slit-system
monochromator
0.8
0.7
sufficiently
the arc does not remain
of vibration; also,
0.6
Intensity of \4348 vs
from arc axis.
Fig. XI-5.
alignment
100 cc-atm/min
0.5
0.4
0.3
0.2
DISTANCE FROM ARC AXIS
A new
50 AMPS
=
CATHODE RADIUS
0.1
0
=
with magnetic
there
800
by dif-
determined
as
seems
gauss
and/or
to
flow
field
is
to be an
Z000
rate
gauss.
are
been made.
the
axis to the
exponential -
At higher pressure it broadens
(XI.
PLASMA PHYSICS)
I
70
50AMPS
F = 100 cc-atm min
65
-
60
-
o
X 4806
0~
o
I = 40 AMPS
70
F
65
=
180 cc-atmmin
-
x
4o
0a~;X
X 4609
zS
60
E
D
60
u
A
55 -
-
I
20 AMPS
F
180cc-atm/min
X4609
55
50
X 3588
X 4348
-
~~
38
45
40
500
6
7
8
9
1000 11
12
13
14
1500 16
17
18
19 2000 21
B (GAUSS)
Fig. XI-6.
Arc width vs magnetic field.
considerably and the exponentials are flatter. Also, we looked for a bump in this crown
which might be due to excitation by primary electrons from the cathode. None was found
within the resolution of the measurements, which is perhaps a few per cent near the
intensity maxima. These observations lend further support to our general analysis,
which assumes that all of the light is generated by plasma electrons rather than by a
monoenergetic beam issuing from the cathode.
b. ii. Intensity versus current measurements have to be repeated with the collimating
slit set on-axis because the alignment problem became worse when these measurements
were made.
Generally speaking, however, the intensity seems, first, to rise and then
to drop off rather sharply as the current is increased. This indicates that the n is
markedly depleted, and n++ is considerably increased near the axis at high current.
C. D. Buntschuh
References
1. C. D. Buntschuh, Quarterly Progress Report No. 65, Research Laboratory of
Electronics, M.I.T., April 15, 1962, pp. 79-87.
2. L. M. Lidsky, S. D. Rothleder, D. J. Rose, S. Yoshikawa, C. Michelson, and
J. R. Mackin, Jr., J. Appl. Phys. 33, 2490 (1962).
3. H. S. W. Massey and E. H. S. Burhop, Electronic and Ionic Impact Phenomena
(Oxford University Press, London, 1952).
QPR No. 70
102
(XI.
4.
T. Holstein, Phys. Rev.
72,
83,
1212 (1947);
PLASMA PHYSICS)
1159 (1950).
5. R. Fowler, Radiation from Low-Pressure Discharges, Handbuch der Physik,
Vol. 22, Edited by S. Fltigge (Springer Verlag, Berlin, 1956), pp. 209-253.
B.
INDIUM ANTIMONIDE PHOTODETECTOR
FOR SUBMILLIMETRIC RADIATION
This photoconductive detector, purchased from Mullard Ltd., is sensitive to radiation
from 100
4 to beyond 1 mm, and has a response time of less than 1 fsec.
The InSb
sample, located at the bottom of a vertical light pipe, is immersed in liquid helium. By
pumping on the helium, the sample temperature is reduced to less than 1. 6°K. The InSb
operates in a 6 kgauss magnetic field obtained from a superconducting niobium solenoid.
The InSb detector was compared with a Golay cell.
radiation from 200
preamplified,
4 to 300 4, chopped at 15 cps.
and then synchronously detected.
Both detectors were exposed to
The signals from each detector were
The signal-to-noise ratio of the InSb
detector was approximately 25 times better than that of the Golay cell.
Further improve-
ment in signal-to-noise ratio has resulted from increasing the chopping frequency to
3100 cps, and from a corresponding decrease in preamplifier noise.
1o.0
6000 GAUSS
3000 GAUSS
I
_
WAVELENGTH (MICRONS)
Fig. XI-7.
QPR No. 70
InSb detector sensitivity as a function of wavelength
magnetic field surrounding InSb sample varied from
3 kgauss to 6 kgauss.
103
(XI.
PLASMA PHYSICS)
The detector exhibited a short wavelength cutoff, as shown in Fig. XI-7, which is
dependent on the magnetic field strength.
R. E.
C.
PLASMA-WAVE
Whitney
COUPLING ANGLE
The dispersion relation for longitudinal plasma waves is
-n
2
+ (
B-n
n
(1)
1.
Here, the plus and minus signs refer to ions and electrons.1 The following definitions
2
1 -3 2
a
1- P
kT
2
have been used: n is the refractive index, A =- 6 , B =
2
2
, 6 - m2' a =
(1-3p
o
2
p
P2'=
Ob
cos
e)
6
mc
ob is the cyclotron frequency, and e is the
cp
W is the plasma frequency,
angle between the propagation vector and the constant magnetic field.
electron particle flux along the k vector,
The ion flux and
F , for the two modes of plasma waves are
related by the expression,
rr\
T
T
S-
(2)
in which the numeral subscripts refer to the modes.
Equation 2 states that if the elec-
trons and ions vibrate in phase for one mode, they will be out of phase for the other.
The angles for which the phase of vibration changes are the resonance angles given by
tan
+
R
2
+ -1,
tan 2 O
=
2
1.
(3)
These results have been derived and discussed elsewhere. 1,2
Equation 1 can be written in a form that displays the coupling of ion and electron
waves:
(n +A+-B+) (n 2 +A -B)
= AA_.
(4)
If the coupling term, A A , is disregarded,
an angle is determined at which the
separated ion and electron waves intersect on a polar plot of phase velocity as a function
of angle of propagation.
One would suspect that this angle would indicate a region of
strong coupling between the waves and therefore have physical significance.
coupling angle,
A+ - B
+
QPR No. 70
ec,
This
is defined as
-1
+ = A_- - B_.
(5)
104
(XI.
PLASMA PHYSICS)
When ion and electron temperatures are equal, Eq. 5 simplifies considerably and
yields
tan 2 0
(6)
=
c
1 +
p_
2
Hence such a coupling angle will exist only between the cyclotron lines of the a ,
plane.
2
2
If we consider the kinetic energy density (W)
caused by motion along the propagation
direction, that is,
W
2
*
=
(7)
Bin2) 8 rr'
2
ZN
where
2
2E
2
the energy density of the longitudinal electric field, and
, and -is
we obtain
make use of Eq. 2,
S6
).
(8)
In the very special case 6+ =
6 _, that is,
for equal electron and ion thermal veloc-
ities, Eq. 8 states that when the ions dominate the longitudinal kinetic energy of one
mode, the electrons will dominate the other.
For this case,
the angle at which the
kinetic energies of a particular mode change from electron- to ion-dominated will be
the coupling angle
ec
In the general case, however, there is no sharp distinction in
the kinetic energy of the modes.
We shall now examine the potential energy density (which has been given by Allis,
Buchsbaum,
and Bers3):
2
4
NN
=
1
2
T /k
where VT is the thermal velocity;
sure, and
N2
6V
- -V
N
=
is
-
P
(9)
6V
2V
1 kT
z
1/2NzmVT,
the pres-
we find the following relation for potential energies:
1.
(10)
)2
QPR No. 70
with P =
the fractional volume change.
Again using Eq. 2,
S-
4
\
rTkF
105
(XI.
PLASMA PHYSICS)
Equation 10 indicates that if the potential energy is ion-dominated for one mode, it
will be electron-dominated for the second mode. Furthermore, if we set += _, we
find from Eqs. 7 and 9 that
n + 2
AB -AB
A
A -A
+A
AB-A
0.
A -A
Equation 4 when written as a quadratic in n
n 4+ n[A+A+-B-B
] + BBB+
B+A_
2.
(11)
is
BA
= 0.
Equating either the constant or middle coefficients of Eqs.
(12)
11 and 12 results in Eq. 5.
Therefore, the potential-energy interchange occurs at the coupling angle. Thus a physical interpretation has been found for the geometrical coupling between the independent
electron and ion plasma waves.
H. R. Radoski
References
1. H. R. Radoski, Dispersion relation for longitudinal plasma waves, Quarterly
Progress Report No. 66, Research Laboratory of Electronics, M. I. T., July 15, 1962,
p. 100.
2. W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas
(The M. I. T. Press, Cambridge, Mass., 1963), Chapter 5.
3. Ibid., Chapter 8, p. 119.
QPR No. 70
106